Sample Space Exploration

Curriculum Reference

This lesson aligns with the New Zealand Curriculum Year 6 Mathematics learning area, focusing on the Probability strand within the Number and Algebra domain. The key curriculum expectations include:

• Engaging in chance-based investigations with equally likely outcomes by anticipating and identifying possible outcomes for an investigative question.
• Generating all possible ways to get each outcome (theoretical approach), recording occurrences from probability experiments.
• Creating data visualisations for possible outcomes.
• Describing what these visualisations show.
• Finding probabilities as fractions.
• Comparing experimental and theoretical probabilities.
• Using appropriate mathematical language such as "sample space," "outcome," and "event."
• Developing critical thinking by agreeing or disagreeing with statements about chance-based situations, justifying responses using evidence.

The lesson supports development of the following key competencies: thinking, using language, symbols and texts, managing self (personalising understanding), and participating and contributing to group discussions.

Evidence for these objectives and teaching considerations appear thoroughly in the Te Mātaiaho Maths 0–8 resource and NZC Maths Phase 2 documents.

---

Lesson Overview

Year Level: 6 Duration: 45 minutes Class Size: 30 students Unit: Understanding Probability Concepts Lesson Number: 3 of 10 Lesson Title: Sample Space Exploration

WALT (We Are Learning To):

• Define what a sample space is
• Create sample spaces for various scenarios

Success Criteria:

• Create a complete sample space for a given event
• Use correct mathematical vocabulary related to probability
• Represent sample spaces in lists or tables

---

Lesson Plan

Materials Needed

• Dice (real or printed images)
• Coins
• Cards or coloured counters
• Sample space templates (fill-in-the-blank grids/ tables)
• Whiteboard and markers
• Visual aids with illustrations of events and sample spaces
• Dyslexia-friendly fonts and clear illustrations on worksheets
• Digital tools (optional) to simulate probability trials

---

Lesson Breakdown

1. Introduction (5 minutes)

• Begin with a brief interactive discussion: "If I toss a coin, what outcomes could happen?" Let students share ideas, guiding toward the concept of a sample space as all possible outcomes.
• Show visuals of simple experiments like tossing one coin, tossing one dice.
• Define sample space explicitly: The set of all possible outcomes of an experiment.

Dyslexia-friendly strategy: Use clear, large fonts and accompanying icons/illustrations for "sample space," "outcome," and "event." Provide vocabulary cards for reference.

---

2. Guided Learning – Creating Sample Spaces (15 minutes)

• Using the whiteboard and physical materials, model several examples:

1. Tossing one coin (sample space = {Heads, Tails})
2. Rolling one six-sided dice (sample space = {1, 2, 3, 4, 5, 6})
3. Drawing a card that is either red or black (sample space = {Red, Black})

• Provide students with templates to record the sample space for each example.
• Ask probing questions: "How many outcomes are there? Can we list them all?"
• Illustrate how sample spaces can be organised in lists or simple tables.

Differentiation:

• Support students who require scaffolding by providing partially completed templates.
• Challenge advanced learners to consider events with combined outcomes, such as tossing two coins or rolling two dice, listing all pairs of outcomes.

---

3. Collaborative Activity – Sample Space Exploration (15 minutes)

• Divide class into small groups (4-5 students).
• Each group receives a different challenge:

1. List the sample space for tossing two coins.
2. List the sample space for rolling two dice.
3. List the sample space for choosing one colour from red, blue, green.
4. List the sample space for drawing one card from a small deck with red, green, yellow cards.

• Encourage students to draw illustrations for each outcome to aid understanding and support dyslexic learners.

• Groups record their sample spaces on templates.

• Teacher circulates to provide support, observe understanding, and prompt deeper thinking, e.g., "Are these all the outcomes? Could anything be missing?"

---

4. Sharing & Discussion (7 minutes)

• Groups share their sample spaces with the class.
• Discuss completeness and correctness.
• Highlight different ways to display sample spaces (lists, tables).
• Connect the sample space to the likelihood of outcomes (introduce vocabulary like "equally likely").
• Reinforce success criteria and summary of WALT.

---

5. Independent Practice & Reflection (3 minutes)

• Individually, students complete a quick task: create a sample space for tossing one coin and rolling one dice together (e.g., (Heads, 1), (Heads, 2) etc.).
• Reflect with a sentence: "Today I learned that a sample space shows all the possible outcomes for an event."

---

Assessment and Feedback

• Formative assessment through group activities and class discussion.
• Collect students' sample space templates to check completeness and understanding.
• Anecdotal notes during teacher observation, especially for students requiring differentiated support.

---

Differentiation Strategies

• Provide sample space templates as written supports for all learners.

• Use visual and tactile materials for kinaesthetic learners.

• For dyslexic learners:

• Use dyslexia-friendly fonts and clear illustrations.

• Break down explanations into small steps with visual cues and repeated reinforcement.

• Allow verbal discussion and drawing instead of writing when appropriate.

• Extension for advanced learners:

• Explore creating sample spaces for two dice and calculate total possible outcomes (e.g., 36 outcomes).

• Introduce simple tree diagrams representing sample spaces visually.

---

Cross-curricular Links and Engagement

• Encourage students to think about probability in games they play or environmental events (sports, weather).
• Discuss real-life examples where knowing all possible outcomes might be important (planning, games, technology).

---

Summary

This lesson introduces students to the fundamental concept of the sample space as a foundation for understanding probability. Using engaging, hands-on activities aligned with the New Zealand Curriculum, students develop the skills to define and create sample spaces in diverse contexts. The inclusion of scaffolds, extensions, and dyslexia-friendly resources ensures all learners can access and benefit from the learning while building computational and critical thinking competencies.